In the manufacture of methanesulfonic acid, an aqueous product which normally contains about 20 to 35 weight percent of water is produced. In order to obtain an anhydrous, (&lt;2 wt. percent water) product, useful, for example, as a reaction medium in the preparation of aromatic peroxy acids where excess water retards the reaction, the water must be removed while minimizing the formation of decomposition products. Of special concern is the need to avoid the formation of methyl methanesulfonate (CH.sub.3 SO.sub.2 OCH.sub.3). This compound is a known carcinogen. A two-step distillative purification process for lower-alkanesulfonic acids is disclosed in U.S. Pat. No. 4,035,242. Water vapor is removed in the first step by distillation and the major portion of the alkanesulfonic acid is vaporized and removed by vacuum distillation in the second step. In the example, a product having a purity of 98.89 weight percent is obtained. Besides water, the product is reported to contain an average of 0.08 weight percent methyl methanesulfonate and 0.46 weight percent sulfuric acid. I have now found a process in which an anhydrous alkanesulfonic acid can be prepared, in a single step, without vaporizing the the acid product, to a purity of at least 99.5 weight percent. The acid contains less than 1.0 ppm of methyl methanesulfonate and no detectable sulfates.